The damping ratio ζ = 1 corresponds to:

• A. A. No oscillation and slow response
• B. B. High oscillation
• C. C. Critical damping
• D. D. Underdamped response

Answer: (C)

The damping ratio ζ = 1 is a key parameter in control systems that indicates the behavior of the system's response. When ζ is equal to 1, the system is said to be critically damped. In this state, the system returns to equilibrium as quickly as possible without oscillating. This is a desirable condition in many engineering applications because it ensures that the system stabilizes quickly after a disturbance without overshooting the desired position.

In the context of damping, if the damping ratio is less than 1, the system is considered underdamped and will oscillate before settling. Conversely, if the damping ratio is greater than 1, the system is overdamped and takes longer to reach equilibrium compared to critical damping. Thus, a damping ratio of exactly 1 defines the critical point where the system is optimally damped to avoid oscillations while ensuring a fast response.